凝聚态和原子物理中的多体现象.doc

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8.514: Many-body phenomena in condensed matter and atomic physics Problem Set #6 Due: 10/21/038.514: 凝聚态和原子物理中的多体现象 问题集 # 6提交日期:10/21/03Bardeen Cooper - Schrieer theory巴丁-库柏-施里弗理论1. Quasiparticles. 准粒子Consider quasiparticles of a BCS superconductor,考虑BCS超导体准粒子where and , etc., are Bogoliubov quasiparticle operators.其中, 和等是巴格寥夫准粒子算符。a) Find out how many particles are contained in one quasiparticle. For that, consider a state with one quasiparticle added to the BCS ground state. 找出一个准粒子中包含的粒子数。为此,考虑将一个准粒子加到BCS基态时的态。evaluate the expectation value估算期望值and express the result in terms of the Bogoliubov anglep. CanNbe nagative? Explain. 并将结果用巴格寥夫角p来表述。N可以取负值?解析该结果。b) Consider momentum and spin of a quasiparticle in the state (2). What are they? Do they depend on the Bogoliubov angle? 考虑态(2)中的准粒子动量及自旋。它们是什么?它们依赖于巴格寥夫角吗?2.Gap equation. 能隙方程For a BCS superconductor derive the gap equation推导BCS超导的间隔方程with E* the interaction cutoff parameter (E* EF for nonretarded contact interaction). 其中E* 是作用切断参数(E* EF 非延迟接触作用)Study the gap as a function of temperature. Show that decreases monotonically with T and vanishes at a certain temperature T = Tc. Find the value Tc.研究关于温度的函数。证明随着T单调减小同时在T = Tc时为零。找出Tc值3. Gap suppression in a superflow. Critical current. 在超流中的能隙压缩。临界流Superflow in a superconductor is described by the order parameter with spatially varying phase, (r)= e2iqr , which is related to the superflow velocity by vs = q/m.起导体中的超流是用随相随空间变化的序参数(r)= e2iqr来描述的,它与超流速通过vs = q/m 联系起来BCS quasiparticles in the presence of superflow are described by the Hamiltonian存在超流的情况BCS 准粒子由以下哈密顿量描述 which can be dioganalized by a Bogoliubov transformation in which the states p + q and p+ q are paired up.它可以通过态 p + q 和 p+ q 成对的巴格寥夫变换来对角化。a) Find the quasiparticle spectrum. Assuming |q| pF , show that the result can be interpreted in terms of Doppler shift Ep= Ep + vsp, where Ep is the spectrum in the absence of the flow. 找出准粒子谱。假设|q| pF,证明结果可以用多谱勒频移Ep= Ep + vsp来解释,其中Ep 没有流动时的谱。b) Show that the energy gap between the BCS ground state and the first excited state is reduced in the presence of the flow. Find the critical velocity at which the gap vanishes. b)证明BCS基态及第一激发态之间的能隙在存在流动的情况下将被缩小。找出能隙消失的临界速度。c) Consider BCS pairing in the frame co-moving with the flow. By using Galilean invariance, or otherwise, argue that the gap equation and thus the order parameter are not affected by the flow. Combined with the result of part b) this shows that the energy gap and pairing amplitude aint necessarily have to be equal. They happen to be equal in a clean superconductor in the absence of external pair-breaking fields or flows, but are not equal in general. c)考虑与流动同时移动的参考系下的BCS配对。利用伽利略不变性,或相反,说明能隙方程及由此所得的序参数 不受流的影响。结合b)的结果,证明能隙及配对幅不需要一定相等。它们只有在没有破坏配对的外场或流的干净超导体中才偶然相等,一般而言它们不相等。
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